!"!"#!$#! %%&%
'( ) *%)% )*"%%)%%
+,"- "...
!
"
#
$%&'
( )
!" #
$ %
&
'
()* !
" (
+
Æ #
# '," '-.! '/ " '$!"
'*")' " 0' !"
# # # 1 2 3 ). !
' " )- !"
5 # # "
4 5 4
6 #7 8 5 #
#" # 5 9 9
#1:+;7 "
1:+ # 4 4
$4 # )* " '' " ') !" '< !
#1:+
4 8
), !
# $
# #
(<! 5" #"
"
" #
# =
6 " > #
# #
" #
#
" " #
#
#? @+$:
# " @+$: 9#
) ! # 7 #
0 A " 7
##= $
" " 9 #
#
9 " 54
# 5 A "
#
8 9
" 4'
54 " $4
: )* !" "
' ' #
6 "
# "
5 9
& )
" 9
9 #
A" 9
;< ! 9
# # 8
9 " 4 9 ##
# B 3 "
# # "
9 #
" 9
A" )" 4 9 " #
# "
9 # 8
#
9 =4#
9 A " 4
)" #
3 9
# 9 )
% " 4 9 #
7
# # #"
# # ; 7 #
#? #"
# 7
7
# 7
; 4 # "3# )<!
7 # ## "
A " C )! ' ! 7
# # $ 1 > )) " )0!
7 # 8
9 $ @ " : "
1" D * ! )E 4 5
7
# $ $4 3# )< !
E F '. !" 0 " , !
$ B 7
/! 7 # #
# #
# " # 7
( " 1:+; $ #
7 7 # # #
" 7 # #
7 # "
#9
#
@+$: ; # #
7 #1:+
4 '< !"
54 # @"
@+$:
"
# # 4 G
5
# !( "
8 9 %
&
H
-!" - #
$ "
"
" ; "
# "5
" >
7 ##? 7
#? $4
#? #
8 # +E1:+% - -"
- I
-
- & I!
0
H
H
I# 8 4
# " # # & - &
&"4
4 4 # +E1:+" 5
8 4 8 9 0 5"
# 5 -Æ!" Æ -
7 # 7 8 "
Æ- Æ 7 8
4 %
HÆ
Æ
HÆ)HÆ
Æ
H
'
HÆHÆ)HÆHÆHÆ
Æ
HÆÆ
)Æ
H
'
HÆHÆÆHÆ
)Æ
-'! 7 44 &-"
4 &'" : B 4 &')
# 5 8 # "
ÆÆ"&')'
&-') '" " 1
ÆÆ
ÆÆ
#
-Æ
ÆÆ
Æ
&-' &' ' ;
I
IÆÆ
8 ÆÆ A"
"
""'
"
I
#
# "+E1:+04# '1:+"
'% &Æ& ' )'-"
-
H
H
Æ H H Æ H
H'
!
H'
! H
# 4 ' # ' '
4 #% &') '"
) H )
&'
& ) )
&H'
- + ' Æ H
4 "
" " 5 "
-'! "
Æ 5
#
"
1:+." &'(
"# 1:+&'Æ"
# 9
+ # "
" # 1:+ . 1:+
7 ' " 5 ' ' 4%
' 7
&-'
&' '%
" 5 ' 8%
#
#7 1:+ 7 '%
- H - /
6 " "# # 45
#" +4"# 5 A
+ 1:+" 4 '- ! >#
5 8 # 7 8
L 8 4
+#" 8 7 " # 8 4
#7 "
+4 " #
"1:+/
# # 5"
# 7 # 8 I
4
&'"
1:+% - H
# 7
B 1:+ 5 #
4 : 1:+" 5 1:+
# %
""' 8
7 "
' #
" ''# 4
# 9 '' 4
" #1:+<"
1:+ 9 7 4 (
1:+
ÆÆ
+4" 1:+ 9
( "
# # 1:+; #
#9 ( '
; 7 #? 4
" #
#
$4 #
#?##
$4 9 #
@"#
# #'-''' #
# #-'''
# # #? #
# @"
#? %
H
"
4 # *!
7 D G
#? #
" #? " #
5 7 #%
7
#?
- H
# %
7
#?
*
9 7
#
7 #, @+$:
7 #" 4 ##"
) ! ? @+$: #
7 # ; #=
@+$:
9 9? "
, FF
FF B1+ " FF
4 @+$: ?
@+$: @+$:
# 7 # 4
# " (
#? 9 * ?
@+$:
7 # # ) !
:"
Æ # "
1:+ 7 # 7 "
1:+ #('- ": <!
"5
# " # $
" 1
H
# 5 1:+< M# 4
( " M# 4
9 4 A # " 5 4
%
%
- & H
%
& - & H
! %
& - " H
# " "
#
#" J#
>K " #$&%&$
#$
&$ &
7 '- " <.!
" J+K
4 " # #
#"#?#
!!
"
(" #
$ % &'
H
& -
-
H
-
& - !
"
H & H
(H ''
" " (
9 5 9
# ,* #?
#
)! * "
+%
9 ' 84
# "
4 " ''
H
(
Æ # " 4
#
" 4 1:+ 7
"" " 41:+'-
9 #
"
4 1:+ '- # # 1:+
#
" # 5 9 '- #
'- " ' !
" * 7
# B" #? "
#? J K" 17
8 # "
# * 5 )/ ! 4
) " /0! E 4 A" JK
# * ; ) !
@+$:"
7 #
! "
; # ; "
" @+
++ @+ # # 7 *(
++ # , E
@+$: # ) ! C" #=
5 # 7
# A"# #
(" # #@+
++ # #
" * 7 #
; # 8 #"
9 9 7 (
@+ # "
1
# * ;
!
!
; #?
#
7 #% 5 4'
"
7
(H
'
#?
H
( &
H
7 H H ' #? H H & - - H -
& - !
') ' # (
7 # # @+$:
') # Æ
> 5 " "
@+# # " %
!" 1 (" ! -' # '
#
5 4 ' 1
# ) &
#$ %"1 # ## 49 7 H H ' #? H H & -
& - !
'0
# 4 " # #
9
& "
#
(
# 4 *
' !" ;
* H* * * * -'! *
4--' (
+
+ * ( "
*
*
!*
H
!
*
: 5 4 '
) # ) H'
3 '
'
# 8
"# = #?
B" @+$: "
4 # 4 " 4 '
'. # --' 7
5 # '"
# #
#
> " 7 ? 4
7
5 # ) ! "
# # 5
B" 4
9 #
6 @+
" ++
,# A
# "0-!
++ # ,%
7
#?
&
Æ &
Æ C ( " Æ
9
++" 4 # 1
&
5 %
H
,
H-
- - # ? A " #
4
++ 9
)%
#
(
( #
-
-(
$
Æ
.&
" .-' "
' "
&
'
J K #
; " ++ #? #
,
? 4 5
- ++ %
" 7" # # @+$:"
"# J K#? A
" 4 ) !
# )"
#
: "
"B 7 9 %
&
9 %
Æ & /&
/ -' 9
5 9
#
++
!" 1 (" !" /
" . # -' 1
'
0 - # 1 1
# 5 4
)' 1
"$ #
) &
#$ %" 49
7 H H 1 #? H H & Æ H H & / & 0 ' #
& "
H # (
# 4 ,
' "12
'# 2'
H )
H'/
"
) !* $" ;
* * * H* * -'! *Æ
* .&
* & * & * * ' * -*****
+ !* , $"
+
+ * &*
( " , * * *, " !* H )
H'/
* * * :/ -/ !
5 4 1
3 ' ) #)H'
@+ ++# (#
8 " "
5 ;
# 4
# 9 # (
4 # 5
9
# " $ !%
; 4 @1
@+ ++; )'''.
7 #
&'" 8 #
I 4 4
A "
7
Æ-"
## " # 1:+.
& -. % & '/" Æ & '" &
" Æ & -') " & -- "
&--)
@+++( "
5
9#? %
' ) Æ- 4 ( Æ
; #? 7 " 4
@+ ++ # # #?
7 "
"
@+$: , 9 " 4
##( "#5
;# #4"
#? # "
@+" ?
4 9 '0"
* 7 * &(
9 '0"
#
" # '0 4
9 # 4 #" '
# 4 5 9 "
# # #4" #
* 1:+ <
*
1:+ ##
" #" 1:+ # #
"
* 1:+
. " # 1:+ .( '- " '<!
4 "
# 7 " 0- !
1:+. "
*
++ 49
' 6 @+" 1:+ @+"
' # # #"
4
@+
'/
4 # ,- @+ #
@+$: ," @+ ,- '/"
#
" #?
@+
#
++
H
'<
"
#++ 4#
@+$:,##
#?
!
# &)%
' ,- -,--0
& , ; &
& -) #
A ++"
# #
#
#( "
. 5 5 7
#@+ ++ A' )"
7 ## B#?
# # ++ # # @+ 9
A ' A ) "#
++
N
0' 0-** -* ')/*.'' '-,0)-)
@+
N
<0 .</-'-. ,<)) -<
++ @+
# "
# "
# # ++ @+ #
A & , & *"
% ' & -." ) &
" 0 &
A0L. JD #K
&-.
OA 5" @+
!
$ # &''%
' 0'- ---.' < *'- '<),<
) .'- --),, , '-'- ).00*
0 '- -'' / * '''- 0))<<
. /'- -0'0' '- ')'- .-**.
<'- -/.*< '' '0'- -''<
/ ,'- '')'<
$4' A
$4'" #
++
N
<0 <-'.,-* /,,-0' )))-/-0
@+
N
2
4
6
8
10
12
14
0
0.2
0.4
0.6
0.8
1
0.2
0.3
0.4
0.5
0.6
0.7
S
PIPA: 2 options
t
σ
(S,t)
A'% $4 '"#++") #
2
4
6
8
10
12
14
0
0.2
0.4
0.6
0.8
1
0.2
0.3
0.4
0.5
0.6
0.7
S
IMPA: 2 options
t
σ
(S,t)
5
6
7
8
9
10
11
12
13
14
15
16
0
0.5
1
1.5
2
2.5
3
S
V
Option values, E = 8, T = 1
obs
IMPA
PIPA
payoff
A 0% $4 '" G++G@+ " $
&,
5
6
7
8
9
10
11
12
13
14
15
16
0
0.5
1
1.5
2
2.5
3
3.5
4
S
V
Option values, E = 8, T = 1
obs
IMPA
PIPA
payoff
A .% $4 '" G++G@+ " $
2
4
6
8
10
12
14
0
0.2
0.4
0.6
0.8
1
0.2
0.3
0.4
0.5
0.6
0.7
S
PIPA: 11 options
t
σ
(S,t)
A % $4)" #++"'' #
2
4
6
8
10
12
14
0
0.2
0.4
0.6
0.8
1
0.2
0.3
0.4
0.5
0.6
0.7
S
IMPA: 11 options
t
σ
(S,t)
<L, "#@+ ++#
!
A # &'.%
' - --)<* , '- -'' /
) /- -').- * /'- -0'0'
0 <- -0,,. '- <'- -/.*<
. ,- -,--0 '' ,'- '')'<
*- '..*/ ') *'- '<),<
/ '-- )'*'0 '0 '-'- ).00*
< ''- 0- 00 '. '''- 0))<<
; # A 4 "
$4'
++
N
// )-)0'-* '0 '<' )/- *-0
@+
N
,- '<-,/-0 ' </.-.
&, &* A''L')
4" #++
# @+ = %
#
&-.
!
&-.##
" # # # (
P
H--<-<
" #? #
&
Æ- P
&
Æ-
--<-<
5
6
7
8
9
10
11
12
13
14
15
16
0
0.5
1
1.5
2
2.5
3
S
V
Option values, E = 8, T = 1
obs
IMPA
PIPA
payoff
A <% $4 )" G++G@+ " $
&,
5
6
7
8
9
10
11
12
13
14
15
16
0
0.5
1
1.5
2
2.5
3
3.5
4
S
V
Option values, E = 8, T = 1
obs
IMPA
PIPA
payoff
A ,% $4 )" G++G@+ " $
2
4
6
8
10
12
14
0
0.2
0.4
0.6
0.8
1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
S
PIPA: 14 options
t
σ
(S,t)
A *% $40" #++"'. #
2
4
6
8
10
12
14
0
0.2
0.4
0.6
0.8
1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
S
IMPA: 14 options
t
σ
(S,t)
5
6
7
8
9
10
11
12
13
14
15
16
0
0.5
1
1.5
2
2.5
3
S
V
Option values, E = 8, T = 1
obs
IMPA
PIPA
payoff
A''% $40" G++G@+ "$
&,
5
6
7
8
9
10
11
12
13
14
15
16
0
0.5
1
1.5
2
2.5
3
3.5
4
S
V
Option values, E = 8, T = 1
obs
IMPA
PIPA
payoff
A')% $40" G++G@+ "$
# #?%
--<-<
34
; $4 'L0 #? >
@+ ++ # 34"
&-)
7# ; 4 ++ ) 4"
# #?
# "
++ @+
34
) ) , *-) ,,<) -< .**, -0
'' ' ..)-) .00-'-0 )<.*)-)
'. '--. -) /**0--. 0.*,*-)
"
; 7 # 4 #
++ @+ # #?
A "
#?
&
H
#? "
# # 5 A 'L'-
% & ' & -') "
# # @1 " #
G >#"5
" # 4 #
JK "#
# 5 ++ @+
# 4" 9 #"
-
# # " #
---'-
4 # " $40
&-) # #
2
4
6
8
10
12
14
0
0.2
0.4
0.6
0.8
1
0.2
0.3
0.4
0.5
0.6
0.7
S
PIPA: 14 options, tightened bounds
t
σ
(S,t)
A '0% $40"#++ #
2
4
6
8
10
12
14
0
0.2
0.4
0.6
0.8
1
0.2
0.3
0.4
0.5
0.6
0.7
S
IMPA: 14 options, tightened bounds
t
σ
(S,t)
++
,) <.,/0'- .-.0'' '-.</-)
@+
N
,- )' 0'-0 '0),--0
" # # 9 #
" 9 #?
" $40 # #
5 #?
'
)
Æ-
H '
)
&'''' &-0 A '
'/
++
N
,) '-*.''- ''0/-'. <,' *-)
@+
N
,- .,<)-' ',.,*-'
OA # " =
# # #
#?
# #? " #4#
&
; "
7 # @+$:
8 @+$:" ++
@+" B #
# ##?
" "
7 #1:+ @
# #
% ?
4 +>3 +
1Q 4 1:+.
2
4
6
8
10
12
14
0
0.2
0.4
0.6
0.8
1
0.2
0.3
0.4
0.5
0.6
0.7
S
PIPA: 14 options, with
σ
obs
= 0.35
t
σ
(S,t)
A ' % $40" #++ &-0
2
4
6
8
10
12
14
0
0.2
0.4
0.6
0.8
1
0.2
0.3
0.4
0.5
0.6
0.7
S
IMPA: 14 options, with
σ
obs
= 0.35
t
σ
(S,t)
4 " # +>3
#
@1 +3: 613 G'.- # ' 7
% Æ & '" Æ & -) " & . & '" & -- "
&--)
: B 5 8 &- ; #+>3 +7
4"' A+" '<
A+>3"
$
H
"
4 *--
+>3 4
" +>3 #5 # 54 ''
J#K A" # 8 ')!
-' # A+>3"
# 9 #" #
# 4 6 #" J48K 5
4 8 &
OA " 74 "+>3
( " 7 4 "
# +>3 "
$-.% :$ @
'(
'! @ "J@ # K"":
@ " B R 6 '**<
)! @ ": A "3C E"J:#
7 K" . '**< 0<L/.
0! B :" JS K" @ + S " "
!"3'**,
'<0L'<,
.! " J
K" " # '*,'0/0L0,'
! A @"J #K""
$,' '*<0 /.<L/ *
/! MB @ M " JB
'/ ,
0) '/
- )
/-
0-,-
.-*- .
'--
-'.-
<-)--
'--0-- '
-.--
)--<-- 0
:+6
0- ',0
0/ )<<
.0 .'0
.< -,
< .
' <
0
,0-/) ')/0
<) )'),
<) )*<.
,- .)<*
''' ,0/'
:+6
'- -''
.- -/0
** )'0
'.. 0<)
)/0 *.'
00/ '0..
.'< ',''
,0, .*')
B
B
B
B
:+6 48
'' '- -'' <,0
0'-'0 )' -00 .'*,
'-' 0 -<< 0.'
-'-' .) '-, )/*/0-*
'/ / '*0 )/-)*-*
'< /, )/) 0
///'-'< <- 0-.
0-0/<'-', '-* /0)
,*,<'-', '<' '/-' 0)..'-*
'* )0, 0)-0
0**,*'-'* 0'< /,< 00. /-*
'")!
B B
¾ !" " #" "
$ %% &
¼
'((
#'% : +>3 "&-- "
&--)"
&'" &- " * &-')
<! D " J # K" % '0
'**< 1''L1'<
,! B:"J K" @ +S ""
!" 3 '**, '<*L',.
*! : " R 1 D" J3 K"
" )'*** <<L'-)
'-! 3; :" M + " 3 " "
+" '**)
''! @CE M+C " JA " 4
4 K" . '**<
'L)-')! @CE M+C "J+ # K"
'0! @CE"M+C " ES3"J1+ 4
' ! E"J+ K" %&'< '**.
',L)-'/! 3 S# " ( ) * &* ( "
@SC"B R '**'
'<! MC M+ " J> K" +
)'**,
0'L/-',! M C" (, , ( ! &" $ " + C"
B M '*,*
'*! M C ;" J K"
" .) '*,<
),'L0--)-! +M"E1# " 1"JD 9
K" )' '**- )/0L),*
)'! C M E " J> K" "
- )) '*,<'.0L' '
))! 3 1 >" J 9 # K"
" ' '**< '0L)
)0! 31 >"J3 8 K"%&' '- '**<<*L,0
).! C1 2 3M3 "J
K" " 0' '*</ 0/*L0,'
) ! TU 1" M + " E 3" $. "
:#6 +":#'**/
)/! M+ "+C B3 ?"J@ 7 17 K"&%
" (*' '**' <L,)
)<! @ 3# "J# K" " /*'**. <<'L,',
),! : D797" J $
K" *< '**, )<0L0,/
)*! +;"MBE C "( +
">4A +">4 '**0
0-! ; "3E:@ M+ " J
9 K" '**/ ' *L'*
0'! 3T "+A" F3D7"J+
